Analysis of Cell Load Coupling for LTE Network

8/11/2019 Analysis of Cell Load Coupling for LTE Network

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SIOMINA AND YUAN: ANALYSIS OF CELL LOAD COUPLING FOR LTE NETWORK PLANNING AND OPTIMIZATION 1

Analysis of Cell Load Coupling for LTE Network

Planning and Optimization

Iana Siomina1 and Di Yuan1,2

1Ericsson Research, Ericsson AB, Sweden

2Department of Science and Technology, Linkoping University, Sweden

Emails: [email protected], [email protected]

Abstract

System-centric modeling and analysis are of key significance in planning and optimizing cellular networks.

In this paper, we provide a mathematical analysis of performance modeling for LTE networks. The system modelcharacterizes the coupling relation between the cell load factors, taking into account non-uniform traffic demand

and interference between the cells with arbitrary network topology. Solving the model enables a network-

wide performance evaluation in resource consumption. We develop and prove both sufficient and necessary

conditions for the feasibility of the load-coupling system, and provide results related to computational aspects

for numerically approaching the solution. The theoretical findings are accompanied with experimental results to

instructively illustrate the application in optimizing LTE network configuration.

Index Terms

3.5G and 4G technologies, cell load coupling, network planning, optimization, system modeling

I. INTRODUCTION

Planning and optimization of LTE network deployment, such as base station (BS) location and

antenna parameter configuration, necessitate modeling and algorithmic approaches for network-level

performance evaluation. Finding the optimal network design and configuration amounts to solving

an optimization problem of combinatorial nature. Toward this end, system modeling admitting rapid

performance assessment in order to facilitate the selection among candidate configuration solutions, of

which the number is typically huge, is essential. In this paper, we provide a rigorous analysis of an

LTE system performance model that works for general network topology and explicitly accounts for

non-uniform traffic demand. The performance model that we study is referred to as the load-coupling

system, to emphasize the fact that the model characterizes the coupling relation between the cells in

their load factors. For each cell, the load factor is defined as the amount of resource consumption in

relation to that is available in the cell. The load value grows with the cell’s traffic demand and the

http://arxiv.org/abs/1201.4116v1





































amount of inter-cell interference. Intuitively, low load means that the network has more than enough

capacity to meet the demand, whilst high load indicates poor performance in terms of congestion and

potential service outage. In the latter case, the network design and configuration solution in question

should be revised, by reconfiguration or adding BS infrastructure. Thus, simple means for evaluating the

cell load for a given candidate design solution is of high importance, particularly because the evaluation

may have to be conducted for a large number of user demand and network configuration scenarios.

The load-coupling model for LTE networks takes the form of a non-trivial system of non-linear equa-

tions. Calculating the solution to the model, or determining solution existence, is not straightforward. In

this paper, we present contributions to characterizing and solving the load-coupling system model. First,

we present a rigorous mathematical analysis of fundamental properties of the system and its solution.

Second, we develop and prove a sufficient and necessary condition for solution existence. Third, we

provide theoretical results that are important for numerically approaching the solution or delivering a

bounding interval. Fourth, we instructively illustrate the application of the system model for optimizing

LTE network configuration.

The remainder of the paper is organized as follows. In Section II we review some related works. The

system model is presented in Section III, and its fundamental properties are discussed in Section IV.

In Section V, we present linear equation systems for the purpose of determining solution existence. In

Section VI, we provide the relation between solving the load-coupling system and convex optimization,

and discuss approximate solutions. The application of the system model and our theoretical results to

LTE network optimization is illustrated in Section VII, and conclusions are given in Section VIII.

II . RELATED WORKS

Planning and performance optimization in cellular networks form a very active line of research in

wireless communications. There are many works on UMTS network planning and optimization. The

research topics range from BS location and coverage planning [3]–[5], [25], [43], antenna parameter

configuration [15], [16], [33], to cell load balancing [18], [34]. For UMTS, the power control mechanism

that links together the cells in resource consumption is an important aspect in performance modeling [2],

[3], [19], [41], [42]. By power control, the transmit power of each link is adjusted to meet a given signal-

to-interference-and-noise ratio (SINR) threshold. By the SINR requirement, the power expenditure of

one cell is a linear function in those of the other cells. As a result, the power control mechanism is

represented by a system of linear equations, which sometimes is referred to as UMTS interference




coupling [15], [16]. Interference coupling can be modeled for both downlink and uplink. For network

planning, the interference coupling system needs to be solved many times for performance evaluation

of different candidate network configurations and multiple or aggregate user demand snapshots. In [26],

it is shown that, for both downlink and uplink radio network planning, the dimension of the power-

control-based system of equations can be reduced from the number of users in the system to the number

of cells. The observation stems from system characteristics that also form the foundation of distributed

power control mechanisms, see, e.g., [19], [42]. In [12], the authors provide theoretical properties

of the power-control-based system, and feasibility conditions in terms of target data rates and QoS

requirements. Motivated by the fact that full-scale dynamic simulation is not computationally affordable

for large networks, the authors of [44] extend the UMTS power-control system by a randomization-based

procedure of service and rate adaptation for HSUPA network planning.

In cellular network planning, the power-control equation system is considered under given SINR

threshold. Thus the system solution and its existence are induced by the (candidate) network config-

uration in question. In a more general context of wireless communications, power control is often a

means in performance optimization, that is, the powers are optimization variables in minimizing or

maximizing objectives representing error probability, utility, QoS, etc., that are all functions of SINR.

There is a vast amount of theoretical analysis and algorithmic approaches for power optimization under

various (typically non-linear) objective functions, where a gain matrix defines interference coupling

[37]–[40]. In [37], the authors identify objective functions admitting a convex formulation of power

optimization, and develop a distributed gradient-projection-based algorithm. Further developments in-

clude algorithmic design utilizing Kuhn-Tucker condition [39], conditional Newton iteration yielding

quadratic convergence [40], and model extension to include explicit SINR-threshold constraints [38].

Another line of research of power control is the characterization of the achievable performance region

under various utility functions and interference functions. The authors of [11] show the strict convexity

of the region for logarithmic functions of SINR. In [7], the authors characterize utility functions and

function transformation of power, for which the resulting power optimization problem is convex. The

investigation in [9] provides conditions under which the boundary points of the region are Pareto-

optimal. In [8], the authors present graph representations of power and interference, and study the

relation between graph structure, irreducibility of the interference coupling matrix, and the convexity

of the utility region.




In contrast to the power-control model, the service requirement of rate-control scheme in cellular

networks is not SINR threshold (or a function of SINR), but the amount of data to be served over

a given time period. Among other advantages, this approach makes it possible to capture the effect

of scheduling without the need of explicitly modeling full details of scheduling algorithms. The rate-

control-based approach is primarily targeting, although not limited to, non-power-controlled systems or

systems with a fixed-rate traffic demand. The approach has been less studied, but is of a high interest

for OFDMA-based networks. In general, the rate-control scheme exhibits non-linear relations between

the cell-coupling elements (in our case, cell loads). The resulting model is therefore more complex

than the power-control model for UMTS. For power control, fundamental solution characterizations

are well-established for linear as well as more general interference functions. For the latter, see, for

example, [10]. For rate-control-based coupling systems (see [23] and Section III), a structural difference

from power control is that, in the former, one element cannot be expressed as a sum of terms, each

being a function denoting the impact of another element, and the coupling is not scale invariant. For

network planning, one known approach is to consider an approximate linear function, obtained from

system-specific adaptive modulation and coding (AMC) parameters, to represent the relation between

date rate and SINR [24], and thereby arrive at a equation system being similar to that of UMTS.

From an engineering standpoint, LTE network optimization is becoming increasingly important. In

[13], the authors provide the fundamental principles of LTE network operation and radio resource

allocation. Among the optimization issues, the research theme of scheduling strategies and radio resource

management (RRM) algorithms has been extensively investigated. See, for example, [6], [20]–[22], [29]–

[31] and the references therein. Two major aspects considered in the references are the balance between

resource efficiency and fairness, and quality of service awareness. In [17], the author gives a survey of

tools enabling service and subscriber differentiation. For cell planning, the propagation modeling, link

budget consideration, and performance parameters have been investigated in [36].

High-level and accurate performance modeling is of high value in planning cellular networks, as full-

scale dynamic simulations are not affordable for large planning scenarios (e.g., [44]). The LTE system

model that we analyze has been introduced by Siomina et al. [32] for studying OFDM network capacity

region with QoS consideration. The work in [32] does not, however, provide a general analysis of the

model, and the major part of the study relies on a simplification assuming uniform load among cells.

In the forthcoming sections, we present both analytical and numerical results without these limitations.




Recently, the authors of [23] have presented a non-linear LTE performance model being very similar

to the one studied in the current paper. That our performance model has been independently proposed

by others supports the modeling approach. The work in [23] provides further an approximation of load

coupling via another non-linear but simpler equation system, along with incorporating continuous user

distribution. Our study differs from [23], as the focus of the current paper is a detailed investigation of

key properties and solution characterization of the load coupling system.

III. THE SYSTEM MODEL

Denote by N = {1, . . . , n} the set of cells in a given network design solution. Without loss of

generality, we assume that each cell has one antenna to simplify notation. The service area is represented

by a grid of pixels or small areas, each being characterized by uniform signal propagation conditions.

The set of pixels is denoted by J . The total power gain between antenna i and pixel j is denoted by

gij. We use J i ⊂ J to denote the serving area of cell i. In a network planning context, both the gain

matrix as well as the cells’ serving areas are determined by BS location and antenna configuration.

For realistic network planning scenarios, the traffic demand is irregularly distributed. Let the user

demand in pixel j be denoted by d j. The demand represents the amount of data to be delivered to

the users located in pixel j within the time interval under consideration. By defining a service-specific

index, the demand parameter and the system model can be extended to multiple types of services (see

[32]). We will, however, consider one service type merely for the sake of compactness.

We use ρi to denote the level of resource consumption in cell i. The entity is also referred to as cell

load. In LTE systems, the cell load can be interpreted as the expected fraction of the time-frequency

resources that are scheduled to deliver data. The network-wise load vector, ρ = (ρ1, ρ2, . . . , ρn)T , plays a

key role in performance modeling. In particular, a well-designed network shall be able to meet the target

demand scenarios without overloading the cells. Hence the load vector forms a natural performance

metric in network configuration (cf. power consumption in UMTS networks). The load of a cell is a

result of the user demands in the pixels in the cell serving area, the channel conditions, as well as the

amount of interference. The last aspect inter-connects the elements in the load vector, as the load of

a cell is determined by the SINRs and the resulting bit rates over the cell’s serving area, and these

values are in turn dependent on the load values of the other cells. To derive the performance model,

we consider the SINR in pixel j ∈ J i defined as follows,




γ j(ρ) = P igij

k∈N\{i}

P kgkjρk + σ2. (1)

In (1), P i is the power spectral density per minimum resource unit in scheduling (in LTE, this

corresponds to a pair of time-consecutive resource blocks), and σ2 is the noise power. By (1), the inter-

cell interference grows by the load factor. In effect, ρk can be interpreted as the probability of receiving

interference originating from cell k on all the sub-carriers of the resource unit. Let B log2(1 + γ j(ρ)) be

a function describing the effective bitrate per resource unit. This formula is shown to be very accurate

for LTE downlink [27]. Thus to serve demand d j in j, dj

B log2(1+γ j (ρ)) resource units are required.

Let K denote the total number of resource units in the frequency-time domain in question, and denote

by ρij the proportion of resource consumption of cell i due to serving the users in j ∈ J i. By these

definitions, we obtain the following equation,

Kρij = d j

B log2(1 + γ j(ρ)). (2)

From (2), it is clear that the load of a cell is a function of the load levels of other cells. Observing

that ρi =

j∈J iρij and putting the equations together lead to the following equation,

ρi = j∈J i ρ

ij = j∈J i

d j

KB log2(1 + γ j(ρ)) = j∈J i

d j

KB log2 1 + P igijk∈N\{i} P kgkjρk+σ

2 . (3)

The equation above represents the coupling relation between cells in their resource consumption.

In vector form, we have ρ = f (ρ, g,d, K , B), where f = (f 1, . . . , f i, . . . , f n)T , and f i, i = 1, . . . , n,

represents the Rn−1+ → R+ function as defined by (3); here, R+ and Rn−1

+ are used to denote the single-

and (n − 1)-dimension space of all real non-negative numbers, respectively. Since in the subsequent

discussions there will be no ambiguity in the input parameters, we use the following compact notation

to denote the non-linear equation system,

ρ = f (ρ). (4)

From (3), three immediate observations follow. First, for all i = 1, . . . , n, the load function f i is

strictly increasing in the load of other cells. Second, for non-zero σ2, this function is strictly positive

when the load values of other cells (and thus interference) are all zeros, i.e., f (0) > 0. Third, the




function is continuous, and at least twice differentiable for ρ ≥ 0.

From the network performance standpoint, the capacity is sufficient to support the traffic demand,

if equation system (4) admits a load vector ρ with 0 ≤ ρi ≤ 1, i ∈ N . In our analysis, however, we

do not restrict ρ to be at most one, in order to avoid any loss of generality. In addition, even if the

solution contains elements being greater than one, the values are of significance in network planning,

because they carry information of the amount of shortage of resource in relation to the demand.

Solving (4) deals with finding a fixed point (aka invariant point) of function f in Rn+, or determines

that such a point does not exist. In the remainder of the paper, we use S as a general notation for the

space of non-negative solutions (fixed points) to systems of equations or inequalities. The system in

question is identified using subscript. Thus, S ρ=f (ρ) denotes the solution space of (4). Note that, for

(4) as well as the linear equation systems to be introduced later, only non-negative solutions are of

interest. Hence, throughout the article, a (linear or non-linear) system is said to be feasible, if there

exists a solution for which non-negativity holds, otherwise the system is said to be infeasible (even if

a solution of negative values exists). The case that (4) is infeasible is denoted by S ρ=f (ρ) = ∅.

A useful optimization formulation in our analysis is the minimization of the total cell load, subject

to the inequality form of (4). The formulation is given below.

mini∈N

ρi (5a)

ρ ≥ f (ρ) (5b)

ρ ∈ Rn+ (5c)

For (5), its solution space S ρ≥f (ρ) is also referred to as the feasible load region. Recall that f (ρ) is

strictly increasing, hence if S ρ≥f (ρ) = ∅, then for any optimal solution to (5), (5b) holds with equality, as

otherwise (5a) can be improved, contradicting that the solution is optimal. In conclusion, any optimum

of (5) is a solution to (4).

We end the section by an illustration of the load-coupling system for two cells in Figure 1. The two

cells have symmetric parameters. In the figure, the two non-linear functions are given by the solid lines.

In the first two cases, system (4) has solutions in R2+, though one of them represents a solution beyond

the network capacity. In the last case, the system is infeasible, as the two curves will never intersect

in the first quadrant. The straight lines with markers in the figure represent linear equations related to

(4). Details of these linear equations are deferred to Section V.




IV. FUNDAMENTAL PROPERTIES

In this section, we present and prove some fundamental properties of the load-coupling system (4).

These theoretical results are of key importance in the study of solution existence and computation.

For compactness, we introduce additional notation to simplify (3) while keeping the essence of the

equation. Define a j = KBdj

, bikj = P kgkjP igij

, and cij = σ2P igij

. These parameters contain, respectively, the

relation between the demand in pixel j and the resource in cell i, the inter-cell coupling in gain between

cells k and i in pixel j , and the channel quality of cell i in relation to noise in pixel j . The load equation

(3) can then be written in the following form,

ρi = f i(ρ), where f i(ρ) = j∈J i

1

a j log2(1 + 1k∈N\{i} bikjρk+cij

). (6)

The first fundamental property of (4) is how fast the load of a cell asymptotically grows in the load

of another cell. We formulate and prove the fact that, in the limit, the first-order partial derivative of

the load function converges to a constant. For any two cells i, k (i = k), ∂ρi∂ρk

is equal to

j∈J i

ln(2)bikja j

1

ln2(1 + 1h∈N\{i} bihjρh+cij(ρ)

)(

h∈N\{i} bihjρh + cij)2(ρ)(1 + 1h∈N\{i} bihjρh+cij(ρ)

). (7)

Theorem 1: limρk→∞

∂f i∂ρk =

j∈J iln(2)

bikja j

Proof: Consider the component for pixel j in the sum in (7), and ignore the constant multiplier

ln(2)bikjaj

. Letting u =

h∈N\{i} bihjρh + cij, (7) can be written as the following expression.

1

u2(1 + 1u

) ln(1 + 1u

) ln(1 + 1u

) =

1

ln(1 + 1u

)u ln(1 + 1u

)u + ln(1 + 1u

)u ln(1 + 1u

)

The theorem follows then from the facts that limu→∞(1 + 1u

)u = e and u is linear in ρk.

By Theorem 1, the load of a cell increases linearly in the load of another cell in the limit, i.e., the

function converges to a line in the high-load region. Moreover, the slope of the line is strictly positive.

The next fundamental property is concavity. The examples in Figure 1 indicate that the load of a

cell is a strictly concave function in the other cell’s load. We show that this is generally true.

Theorem 2: For any cell i ∈ N , f i is strictly concave for (ρi, . . . , ρi−1, ρi+1, . . . , ρn) ∈ Rn−1+ .

Proof: Without loss of generality, consider f n and its (n − 1) × (n − 1) Hessian matrix. Let

u =

h∈N\{n} bnhjρh + cnj. For two cells k and h, the Hessian element has the following expression.




∂ 2f n∂ρk∂ρh

= ln(2) j∈J n

bnkjbnhja j

· ln(1 + 1

u)[2 − (2u + 1) ln(1 + 1

u)]

[ln2(1 + 1u

)(u2 + u)]2 (8)

Let q (u) = 2 − (2u + 1) ln(1 + 1u

). We show that q (u) < 0 for u > 0. This holds, for example,

for u = 1. Next, limu→∞ q (u) = 2 − limu→∞ ln (1 + 1

u)u

(1 + 1

u)u

(1 + 1

u) = 0. Consider q ′

(u) andq ′′(u): q ′(u) = −2 ln

1 + 1

u

+ 1

u + 1

u+1, and q ′′(u) = −1

u2(u+1)2 < 0, ∀u > 0. Therefore q ′(u) is strictly

decreasing and limu→∞ q ′(u) = 0. Hence, q ′(u) > 0, meaning that q (u) is strictly increasing for u > 0.

This, together with q (1) < 0 and limu→∞ q (u) = 0, prove that q (u) < 0, ∀u > 0. By the definition

of u, for ρ ≥ 0, u ≥ cnj which a strictly positive number for non-zero noise power. Hence ( 8) is

well-defined and negative for all ρ ≥ 0. Next, observe that the Hessian matrix is the result of the

following expression.

j∈J i

(bn1 j, . . . , bn(n−1) j)(bn1 j , . . . , bn(n−1) j)T

a j·

ln(2) ln(1 + 1u)q (u)

[ln2(1 + 1u

)(u2 + u)]2 (9)

Because of the form of (9) and that q (u) < 0, ∀u ≥ cnj > 0, the Hessian matrix is negative definite

for any ρ ≥ 0. Hence the conclusion.

From the concavity result, it follows that, for any cell i, f i(ρ1, . . . , ρi−1, ρi+1, . . . , ρn) − ρi exhibits

a strict radially quasiconcave structure. A function is radially quasiconcave, if for a given stationary

positive point, which is in our case a solution to f i(ρ1, . . . , ρi−1, ρi+1, . . . , ρn) = ρi, and any scalar

in range (0, 1), the function value of the scaled point is greater than or equal to zero. If the value is

positive, the function is strictly radially quasiconcave.

Corollary 3: For each i ∈ N , f i(ρ1, . . . , ρi−1, ρi+1, . . . , ρn) − ρi is strictly radially quasiconcave, i.e.,

if f i(ρ1, . . . , ρi−1, ρi+1, . . . , ρn) = ρi, then f i(λρ1, . . . , λ ρi−1, λρi+1, . . . , λ ρn) > λρi for any λ ∈ (0, 1).

Proof: Note that f i(λ(ρ1, . . . , ρi−1, ρi+1, . . . , ρn)) = f i(λ(ρ1, . . . , ρi−1, ρi+1, . . . , ρn)+0(1 − λ). By

Theorem 2, we have f i(λ(ρ1, . . . , ρi−1, ρi+1, . . . , ρn)) > λf i(ρ1, . . . , ρi−1, ρi+1, . . . , ρn) + (1 − λ)f i(0).

Since f i((ρ

1, . . . , ρ

i−1, ρ

i+1, . . . , ρ

n)) = ρ

i and f

i(0) > 0, the result follows.

In a real-life LTE network, if the capacity is sufficient to accommodate the demand, then the network

load will be at a stable working point, which should be unique. Thus the performance model f is

reasonable only if uniqueness holds mathematically. The following theorem states this is indeed the

case. In the rest of the paper, the unique solution, if it exists, is denoted by ρ∗.

Theorem 4: If S f (ρ)=ρ = ∅, then it is a singleton, i.e., f (ρ) = ρ has at most one solution ρ∗ in Rn+.




Proof: Suppose there are two solutions ρ1 and ρ2, both satisfying (4), and ρ1 = ρ2. Let m ∈

argmini=1,...,nρ1i /ρ2

i , and λ = ρ1m/ρ2

m. Thus ρ1m = λρ2

m. Assume λ < 1. Then by construction, λρ2 ≤ ρ1,

and because f is strictly increasing in the domain of Rn+, f m(λρ2) ≤ f m(ρ1). Also, by Lemma 3,

λρ2m < f m(λρ2), and thus f m(ρ1) > λρ2

m. Note that f m(ρ1) = ρ1m = λρ2

m gives an contradiction.

Therefore λ > 1. Considering scaling down ρ1 with λ instead, and applying the same line of argument,

a similar contradiction is obtained. Hence the conclusion.

V. DETERMINING SOLUTION EXISTENCE AND L OWER BOUNDING

Having proven solution uniqueness, we examine the existence of ρ∗, that is, whether or not (4) has a

fixed point. There are a number of theorems characterizing the existence of a fixed point (e.g., Brouwer’s

fixed-point theorem in topology). However, these results do not apply to (4) because, in general, the

output of function f is not confined to a compact set in R

n

+. In this section, we use a linear equation

system for analyzing solution existence. To this end, we first present and prove some basic properties

of the optimization formulation (5).

Theorem 5: Assume S ρ≥f (ρ) = ∅, i.e., there exists ρ ≥ f (ρ) > 0, then (5) has an optimal solution.

Proof: Consider the optimization problem min

i∈N ρi,ρ ∈ S , where S = S ρ≥f (ρ) ∩ {ρ ≤ ρ}.

By the assumption in the theorem, S = ∅. From the definition of S , it is clear that any point being

arbitrarily close to S (i.e., boundary point) is in the set, thus S is closed. In addition, S is bounded

since S ⊆ Rn+ ∩ {ρ ≤ ρ}. Hence S is compact, and the result follows from Weierstrass theorem in

optimization.

Corollary 6: If there exists ρ ≥ f (ρ) > 0, then S ρ=f (ρ) = ∅.

Proof: Follows immediately from Theorem 5 and the previously made observation that any optimal

solution to (5) satisfies (5b) with equality.

To further characterize solution existence, we define the following type of linear equation systems,

ρ = h(ρ) = H · (ρ − ρ) + f ( ρ), (10)

where h = (h1, . . . , hn) is a vector of linear functions, and each of them is defined in Rn−1+ → R+,

ρ is a vector in Rn+ with given values, and f ( ρ) is a vector-function with elements defined by (6). In

(10), H is an n × n matrix where the diagonal elements, H ii, i = 1, . . . , n are zeros, and the other

elements H ik, i = k, are strictly positive. Note that if H is the Jacobian of function f evaluated at

point ρ, (10) is a linearization of the non-linear equation system (4) where the right-hand side of




(10) represents the tangent hyperplane to function f (ρ) at ρ. Such linear approximations are further

discussed in Section VI.

Observing the fact that the partial derivative (7) asymptotically approaches a constant, as formulated

in Theorem 1, we consider linear approximation of f by means of the linear function having the limit

values of the partial derivatives as the matrix elements in H , and passing through the point defined by

the load function values with zero load. Define h0 the case of h where H ik = ln(2)

j∈J ibikj/a j for

k = i, and ρ = 0. For this linear approximation, there are similarities between the elements of H and

the UMTS interference-coupling matrix (see, e.g., [15], [28]) in that both capture the relation between

gain factors of the serving and interfering cells; however, the target QoS in the interference-coupling

matrix is link quality, whilst in H it is given by the amount of user traffic demand.

If S ρ=h0(ρ) = ∅, the solution, denoted by ρ0h, is clearly unique. The lemma below states that the

linear function h0 provides an under-estimation of the true load function f , thus ρ0h, if exists, gives a

lower bound on the solution to the non-linear system (4).

Lemma 7: h0(ρ) ≤ f (ρ) for any ρ ≥ 0.

Proof: We prove the validity of the result for an arbitrary cell i, that is, h0i (ρ) ≤ f i(ρ), ρ ≥ 0.

Because both h0i (ρ) and f i(ρ) are formed by a sum over j ∈ J i, it is sufficient to establish the inequality

for any j ∈ J i. Let u =

k∈N\{i} bikjρk+cij. The proof boils down to showing the following inequality.

1

log2(1 + 1u

) − (u − cij) ln(2) = ln(2)

1

ln(1 + 1u

) − (u − cij)

≥

1

log2(1 + 1cij

) =

ln(2)

ln(1 + 1cij

)

Note that u ≥ cij by definition. The inequality holds as equality for u = cij. It is then sufficient to

prove that 1ln(1+ 1

u) − (u − cij) is increasing for u ≥ cij. Taking the derivative and doing some simple

manipulations, one can conclude that the derivative is non-negative corresponds to the inequality below.

q (u) = u(u + 1)ln2

1 + 1

u ≤ 1, u ≥ cij (11)

One can show easily that limu→0+ q (u) = 0, hence (11) is satisfied for some u ≤ cij. Moreover,

limu→∞ q (u) = limu→∞ ln(1 + 1u

)u ln(1 + 1u

)u + ln(1 + 1u

)u ln(1 + 1u

) = 1. Hence it suffices to prove

that q ′(u) = (2u + 1) ln2

1 + 1u

− 2 ln

1 + 1

u

≥ 0, u ≥ 0. Using the fact that ln(1 + 1

u) > 0 for all

u > 0, the non-negativity of q ′(u) for u ≥ 0 becomes equivalent to that the second numerator in (8) is

negative, which is proven in the proof of Theorem 2, and the result follows.




From Lemma 7, one can expect that the load-coupling system (4) has a solution, only if a solution

exists to ρ = h0(ρ). The following theorem formalizes this necessary condition, and establishes the

result that ρ0h bounds ρ∗ from below.

Theorem 8: If S ρ=f (ρ) = ∅, then S ρ=h0(ρ) = ∅ and ρ0h ≤ ρ∗.

Proof: Consider the following linear programming (LP) formulation.

mini∈N

ρi (12a)

ρ ≥ h0(ρ) (12b)

ρ ∈ Rn+ (12c)

Similar to the result in Theorem 5, it can be easily proven that (12) has an optimal solution if there

exists any ρ ≥ 0 satisfying (12b). In addition, it is clear that any optimum is in S ρ=h0(ρ), and S ρ=h0(ρ)

is either empty or a singleton. Consider ρ∗. By Lemma 7, h0(ρ∗) ≤ f (ρ∗) = ρ∗. Hence ρ∗ is a feasible

solution to (12). It follows then S ρ=h0(ρ) = ∅. Furthermore, (12) obviously remains feasible with the

additional constraint ρ ≤ ρ∗. Since the LP optimum is unique and equal to ρ0h, ρ0

h ≤ ρ∗.

By Theorem 8, the linear system ρ = h0(ρ) is potentially useful for detecting infeasibility. If the

linear system is infeasible, then it is not meaningful to attempt to solve (4). In addition, if feasibility

holds for ρ = h

0

(ρ), the solution provides a lower bound to the true load values. Thus having ρ

0

h

close to one indicates an overloaded network, and its corresponding configuration can be discarded

from further consideration in network planning, without the need of solving the non-linear system (4).

In Figure 1, the lines with markers represent the linear function h0. In the first two cases, ρ∗ exists,

and solving the linear system leads to a lower bound ρ0h (i.e., the intersection point of the lines) of ρ∗.

In the last case, the linear system has no solution, and consequently S ρ=f (ρ) = ∅.

Thus far, it has become clear that ρ = h0(ρ) provides an optimistic view of the cell load. We are able

to prove a slightly unexpected but much stronger result. The linear equations ρ = h0

(ρ), in fact, give

an exact characterization of solution existence of the load-coupling system. Namely, that ρ = h0(ρ)

has a solution is not only a necessary, but also a sufficient condition for the feasibility of (4).

The intuition of the sufficiency result is as follows. Consider Figures 1(a)-1(b), for which the linear

equation system has solution. Suppose the slopes of the lines are increased slightly. Intuitively, if the

increase is sufficiently small, the new linear system will remain feasible. Also, the figure gives the hint




that the modified linear function will eventually go above the non-linear load function for large load,

indicating S ρ=f (ρ) = ∅. To rigorously prove the result, we define the linear equation system ρ = hǫ(ρ),

obtained by increasing the slope coefficients of h0 by a positive constant ǫ. That is, hǫ denotes the

case of (10) where H ik = ln(2)

j∈J i

bikj/a j + ǫ,

ρ = 0.

Lemma 9: If S ρ=h0(ρ) = ∅, i.e., ρ0h exists, then there exists ǫ > 0 such that S ρ=hǫ(ρ) = ∅.

Proof: First, note that S ρ≥h0(ρ) has a non-empty interior. In particular, it is easily verified that

λρ0h is an interior point for any λ > 1. Denote by ρ such a point, that is, ρi > h0

i (ρ), i ∈ N .

Letting ǫi = ρi−h0i (ρ)

k∈N\{i} ρk, ρi = h0

i (ρ) + ǫi

k∈N\{i} ρk, i ∈ N . Next, set ǫ = mini∈N ǫi. Then ρi ≥

h0i (ρ) + ǫ

k∈N\{i} ρk, i ∈ N . Thus for this value of ǫ, ρ ∈ S ρ≥hǫ(ρ), and the result follows.

Lemma 10: Consider any ρ > 0 and any ǫ > 0. Denote by λ a positive number. For any i ∈ N ,

limλ→∞

[hǫi(λρ) − f i(λρ)] = ∞.

Proof: Consider the definitions of hǫi(λρ) and f i(λρ). After some straightforward re-writing and

ignoring the constant term f (0) in hǫi(λρ), the difference between the two functions has the following

form.

j∈J i

ln(2)

a j

k∈N\{i}

bikj ρkλ − 1

ln

1 + 1k∈N\{i} bikj ρkλ+cij

+

k∈N\{i}

j∈J i

ln(2)bikja j

ρk

λ (13)

Let q (λ) denote the expression in the square brackets of (13). By repeatedly using l’Hopital’s rule,

one can show that limλ→∞ q (λ) = −12 − cij, which is a constant. Observing that the last term in ( 13)

grows linearly in λ, the lemma follows.

Theorem 11: If S ρ=h0(ρ) = ∅, then S ρ=f (ρ) = ∅.

Proof: By Lemma 9, there exists ǫ > 0 and ρǫh satisfying ρǫh = hǫ(ρǫh). It is easily verified

that λρǫh ≥ hǫ(λρǫh), λ ≥ 1. Using Lemma 10, there exists λ such that λρǫh ≥ hǫ(λρǫh) ≥ f (λρǫh).

Therefore S ρ≥f (ρ) = ∅, and the result follows from Corollary 6.

Theorems 8 and 11 together provide a complete answer to the solution existence of LTE load coupling,

that is, whether or not the system has a fixed point in Rn+ is equivalent to the feasibility of the linear

equation system ρ = h0(ρ). Clearly, given an LTE network design, this feasibility check should be

performed first, before determining the load values. Furthermore, from Theorem 8, violating ρ0h ≤ 1 is

a simple indication of that ρ∗ is beyond the network capacity. For a two-cell example, the solution to




the linear system ρ = h0(ρ) is

ρ1 = f 1(0) + f 2(0) · H 12

1 − H 21H 12, (14a)

ρ2 = f 2(0) + f 1(0) · H 21

1 − H 21H 12. (14b)

With (14), a feasible solution exists when 1 −H 21H 12 > 0, i.e., H 12 = 1H 21

forms the (open) boundary

of the feasibility region in the two coefficients. Note that H 12 and H 21 are linear in the traffic demands

to be satisfied in cell 1 and cell 2, respectively. The derived relation representing the resource sharing

trade-off for the two neighbor cells in this example is well in line with the commonly known radio

resource sharing and capacity region concepts.

V I. CONVEX OPTIMIZATION AND UPPER BOUNDING

Provided that S ρ=f (ρ) = ∅, a solution algorithm needs to be applied to find ρ∗. Solving ρ = f (ρ) is

equivalent to finding the (unique) root of the n-dimensional function ρ − f (ρ). Thus one approach is

to use the Newton-Raphson method. In this section, we show that approaching ρ∗ can alternatively be

viewed as the convex optimization problem formulated below.

maxi∈N

ρi (15a)

ρ − f (ρ) ≤ 0 (15b)

ρ ∈ Rn+ (15c)

Corollary 12: Formulation (15) is a convex optimization problem, and if S ρ=f (ρ) = ∅, then ρ∗ is the

unique optimum to (15).

Proof: Because f (ρ) is concave (Theorem 2), ρ−f (ρ) is convex in Rn+. Thus ρ−f (ρ) ≤ 0 is a

convex set. The proof is complete by observing that, similar to (5), optimum to (15) must satisfy (15b)

with equality, and ρ∗

is the unique solution to ρ = f (ρ).

Following Corollary 12, any convex optimization solver can be used to approach ρ∗. In network

planning, one will need to solve (4) repeatedly to evaluate many candidate BS location and antenna

configurations. Typically, the performance evaluation does not have to be exact in order to relate the

quality of a candidate solution to that of another. Utilizing the structure of f , we can numerically obtain

upper bounds to ρ∗ via linear equations. Consider any ρ ∈ Rn+. Using the partial derivatives of f at ρ,




and the point f (ρ), we obtain an upper approximation of f due to concavity. Formally, denote by h

the linear function of (10) where ρ = ρ and H ik, defined by (7), takes the following value,

H ik = ∂f i∂ρk

(ρ) = j∈J i

ln(2)bikja j

1

ln2

1 + 1

h∈N\{i} bihj ρh+cijh∈N\{i} bihjρh + cij

2

1 + 1

h∈N\{i} bihj ρh+cij

The positive-valued solution to the linear system ρ = h(ρ), if it exists, is denoted by ρh. As

established below, h and ρh yield upper estimations of f and ρ∗, respectively.

Corollary 13: h(ρ) ≥ f (ρ),ρ ≥ 0.

Proof: Follows immediately from the concavity of f and the definition of h.

Theorem 14: If S ρ=h(ρ) = ∅, then ρh ≥ ρ∗.

Proof: Consider the linear programming (LP) formulation max{

i∈N ρi : ρ ≤ h(ρ),ρ ∈ Rn+}.

Similar to (15), it is easily realized that the LP formulation, if feasible, has a unique optimum satisfying

ρ = h(ρ). Hence the unique optimum is ρh. Moreover, ρ∗ is, by Corollary 13, a feasible point to the

LP, and hence the LP remains feasible after including ρ ≥ ρ∗, and the result follows.

The process of solving the load-coupling system, e.g., an interior point method for (15), will typically

generate a sequence of iterations approaching ρ∗ from below. By Theorem 14, the iterations can be

used to compute upper bounds, thus yielding an interval confining ρ∗. In order to speed up the process

of network optimization, performance evaluation of a candidate planning solution can use a threshold

of the maximum size of the interval, instead of computing the exact solution of the load vector.

Computing an upper bound ρh, involves solving a system of n linear equations. The same amount

of computation applies to the feasibility check and computing lower bounding ρ0h in Section V. It

is straightforward to see that calculating the coefficients is of complexity O(n2). Thus the overall

complexity lies in the matrix inversion operation that runs in the time range O(n2.3727) and O(n3),

where the former is attainable only asymptotically by the Coppersmith–Winograd algorithm.

VII. NUMERICAL RESULTS

In this section, we numerically investigate the theoretical findings in the previous sections. An

illustrative simulation study has been conducted for a three-site 3GPP LTE network with an inter-

site distance of 500 m, adopting a wrap-around technique. The simulated system operates at 2 GHz

with 10 MHz bandwidth. Each site is equipped with a three-sector downtilted directional antenna with

14 dBi antenna gain. The propagation environment and user distribution follow the 3GPP specification




in [1], assuming propagation model 1 (Okumura-Hata, urban, 8 dB standard deviation shadow fading)

and user generation scenario 4b with one hotspot of 40 m radius per macro cell area. Note that, as

for any system model, the complete assessment of the model validity would also include validating of

numerical results against results from real deployments, which is beyond the scope of the current paper.

The network layout we have used is illustrated in Figure 2. Two layers of users are generated, with

30 users per macro cell area in total, out of which 2/3 (the dot markers) is in a randomly placed hotspot,

and 1/3 (the x-markers) are distributed randomly and uniformly over the area. Each user equipment has

an omni-directional antenna with 0 dBi antenna gain. The traffic demand corresponds to 400 kbit/s for

all users within a duration of one second in the time domain.

Two network configurations are illustrated in Figure 2, with the only difference being the antenna

direction of cell 1, which impacts the sets of users served by the cell and its neighbors. Intuitively,

configuration two is inferior, since it results in that the hotspot users in cell 1’s original coverage area

(see Figure 2(a)) is to be served by cell 8 and/or cell 9, although these users are relatively far away

from the two cells. The likely impact is poorer link quality for the users in the handed-over hotspot as

well as increased number of users to be served by the neighbors of cell 1. These effects are expected

to be seen in the load of the neighbor cells.

First, for both configurations, the existence of system solutions has been verified by finding ρ0h to

the corresponding linear system, as described in Section V. Next, the non-linear coupling system (4) is

solved using the non-linear optimization toolbox of MATLAB. Both ρ0h and ρ∗ are shown in Figure 3.

As expected, the load of cells 8 and 9 increases for the second configuration. At the same time, the load

of cell 1 does not decrease either, even though it serves fewer users under the second configuration.

This is due to a joint effect of several factors. Firstly, as can be seen from Figure 2(b), users served

by cell 1 are likely to experience high interference from cell 6 and vice versa. From Figure 3(b), we

observe that the load of cell 6 has also slightly increased. Secondly, the increased load in cells 8 and 9

implies more frequent transmissions in these cells and thus higher probability of interference to other

cells; this in turn increases the load of the other cells, which can also be clearly seen in Figure 3. We

further note that the solution ρ∗ and thus the second configuration are not feasible from the capacity

point of view.

In Figure 3, we also illustrate the load solutions to the linear systems described in Section VI, i.e.,

the upper bound ρh assuming ρ = ρ0h. Table I provides further details of the quality of both the lower




and upper bounds obtained for all 9 cells in Configuration 1 for which the solution is illustrated in

Figure 2(a).

The tight upper bound indicates the efficiency of the linear approximation described in Section VI.

In average, the estimation is deviates only a few percent from the true load value. For ρ0h

the values

are significantly lower than ρ∗, as ρ = h0(ρ) represents a very optimistic view of load coupling. The

observation sheds further light on the importance of fundamental characterization that the two systems

are completely equivalent in solution existence, despite the large difference in numerical values of ρ0h

and ρ∗. Improving the lower bounds, although being beyond the scope of the current paper, is an

interesting topic for future investigation. For example, in [35], the model discussed in this paper is

applied for load balancing, for which very tight lower and upper bounds are obtained using few fixed-

point iterations. It should be further noted that although the results have been presented for downlink,

the model and the theoretical findings can be also applied to uplink.

For the two network configurations, in Figure 4 we illustrate the behavior of the cell-load coupling

with respect to demand, which is successively scaled up uniformly over the service area. Figure 4(a)

and Figure 4(b) show, respectively, the results for the nonlinear load-coupling system (4), and the linear

equation system ρ = h0(ρ) that provides lower estimation and characterizes feasibility of (4). The two

configurations are distinguished by using respectively solid and dotted curves for the load solutions of

nine cells. In Figure 4(a), the thicker curves represent the load of cell 8. For the linear system, only

the maximum value among the cells is shown in Figure 4(b) for the sake of clarity.

From Figure 4(a), it is apparent that the solution values of (4) grow rapidly in the high-demand region,

and becomes infeasible beyond some point. The feasibility boundaries for the two configurations are

shown by the vertical lines. Configuration one is clearly superior, as its load values, shown by the solid

curves, are below those of configuration two, and the feasibility boundary is considerably higher. For

configuration two, cell 8 has the highest load (the dotted thick line). Using configuration one, some

of the users are served by cell 1 instead (see Figure 2), leading to lower load in cell 8 (the solid

thick line). Note that, for both configurations, when getting somewhat close to the infeasible region,

the solver gives solutions containing some zero elements, indicating the solution is invalid (the system

becomes unstable), before all the values abruptly drop to zeros showing infeasibility. When this occurs,

the distance to the feasibility boundary is, in fact, significant. The behavior shows the importance of

our analysis of characterizing feasibility exactly by the linear system ρ = h0(ρ).




In Figure 4(b), the linear system gives values growing consistently in demand. The system gives the

feasibility boundary point of the non-linear load coupling equations, when the determinant of I − H ,

where I is the identity matrix and H is the matrix defined for h0, equals zero. After passing the

boundary point, the linear system returns negative (infeasible) solutions. Hence the numerical results

verify that the significance of the linear system to identifying solution existence. Moreover, using the

linear system, one is able to conclude, as shown in Figure 4, that configuration one is clearly superior

to configuration two.

VIII. CONCLUSIONS

We have provided a theoretical analysis of the LTE load coupling system originally presented in

[32] and have derived its fundamental properties, including concavity, behavior in limit, and solution

uniqueness. We have also formulated the necessary and sufficient condition for solution existence, The

analysis leads to a simple means for determining feasibility. In addition, we have presented two linear

approximations. The analysis has been supported by theoretical proofs and numerical experiments and

can serve as a fundamental basis for developing radio network planning and optimization strategies for

LTE. Furthermore, the presented linearizations and the bounding-based optimization can potentially be

used for more general convex optimization problems with similar properties.

ACKNOWLEDGMENTS

The work of the second author is supported by the Swedish Foundation of Strategic Research (SSF).

The authors would also like to thank the reviewers for their comments and suggestions.

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0 1 2 30

0.5

1

1.5

2

2.5

3

ρ1

ρ2

(a) Feasible solution within network ca-

pacity: 0 ≤ ρ ≤ 1.

0 1 2 30

0.5

1

1.5

2

2.5

3

ρ1

ρ2

(b) Feasible solution beyond network ca-

pacity: ∃(i ∈ N )|ρi > 1.

0 1 2 30

0.5

1

1.5

2

2.5

3

ρ1

ρ2

(c) Infeasible system: S ρ=f (ρ) = ∅

Fig. 1. An illustration of the load-coupling system of two cells.

−500 −400 −300 −200 −100 0 100 200 300 400 500−500

−400

−300

−200

−100

0

100

200

300

400

500

Distance, [m]

D i s t a n c e ,

[ m ]

Non−hotspot users

Hotspot users

Users served by cell 1

cell 8

cell 9

cell 7

cell 6

cell 5

cell 4

cell 2

cell 1

cell 3

(a) Configuration 1.

−500 −400 −300 −200 −100 0 100 200 300 400 500−500

−400

−300

−200

−100

0

100

200

300

400

500

Distance, [m]

D i s t a n c e ,

[ m ]

Non−hotspot users

Hotspot users

Users served by cell 1

cell 4

cell 5

cell 6

cell 7

cell 8

cell 9

cell 2

cell 3

cell 1

(b) Configuration 2.

Fig. 2. Network configurations for numerical studies.

TABLE I

LOWER AND UPPER BOUNDS QUALITY FOR C ONFIGURATION 1

Bound Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9Upper bound,

|ρ0h−ρ∗|ρ∗ · 100% 5.19 5.55 6.66 4.61 4.93 4.94 3.33 5.20 4.17

Lower bound, |ρ

h−ρ∗|ρ∗ · 100% 48.05 48.15 44.44 43.08 43.21 50.62 51.66 46.82 50.00




1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Cell ID

C e l l l o a d , ρ i

Solution to non−linear system, ρ*

Limit−based lower bound, ρh

0

An upper bound, ρh

(a) Configuration 1: Feasible solution within network capacity.

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Cell ID

C e l l l o a d , ρ i

Solution to non−linear system, ρ*

Limit−based lower bound, ρh

0

An upper bound, ρh

(b) Configuration 2: Feasible solution beyond network capacity.

Fig. 3. Load solutions for the two example network configurations.

0 100 200 300 400 500 600 700 800 9000

1

2

3

4

5

6

7

8

9

10

11

Traffic demand, [kbps]

N o n − l i n e a r s y s t e m s

o l u t i o n , ρ * i

Conf 1: cell 8 load, ρ*8

Conf 2: cell 8 load, ρ*8

Conf 1: feasibility boundary


(a) Growth of cell load with respect to demand for the load

coupling system (4).

0 100 200 300 400 500 600 700 800 900−100

−80

−60

−40

−20

0

20

40

60

80

Traffic demand, [kbps]

L i n e a r s y s t e m s

o l u t i o n , m a x { ρ i }

Conf 1: cell 8, ρ8

Conf 2: cell 8, ρ8



(b) Growth of cell load with respect to demand for the linear

system ρ = h0(ρ).

Fig. 4. System solution with respect to traffic demand.

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Analysis of Cell Load Coupling for LTE Network